The Klauder-Daubechies Construction of the Phase Space Path Integral and the Harmonic Oscillator

TL;DR

This paper explores the Klauder-Daubechies phase space path integral approach, applying it explicitly to the harmonic oscillator to demonstrate its features and potential advantages in quantum dynamics and field theory contexts.

Contribution

It provides an explicit solution of the Klauder-Daubechies phase space path integral for the harmonic oscillator, highlighting its novel regularisation and physical time scale implications.

Findings

Explicit solution for harmonic oscillator case

Demonstration of the regularisation parameter's role

Potential applications to quantum gravity and field theory

Abstract

The canonical operator quantisation formulation corresponding to the Klauder-Daubechies construction of the phase space path integral is considered. This formulation is explicitly applied and solved in the case of the harmonic oscillator, thereby illustrating in a manner complementary to Klauder and Daubechies' original work some of the promising features offered by their construction of a quantum dynamics. The Klauder-Daubechies functional integral involves a regularisation parameter eventually taken to vanish, which defines a new physical time scale. When extrapolated to the field theory context, besides providing a new regularisation of short distance divergences, keeping a finite value for that time scale offers some tantalising prospects when it comes to strong gravitational quantum systems.